An indoor tennis centre has five courts which are booked by members of the public, usually over the telephone.

It has been found that demand for courts in each off-peak hour, i.e. between 9am and 4pm, has a Poisson distribution with a mean of 4.

(a) What is the probability that none of the courts are booked between 10am and 11am on a particular day?

(b) What is the probability that all five courts are booked between 10am and 11am on a particular day?

(c) What is the probability that none of the courts are booked for exactly three of the off-peak hours on a particular day?

(d) What is the probability that all 5 courts are booked for at least 5 of the off-peak hours?

(e) Demand for each hour in the peak-period hours, i.e. 4pm to 11pm, has been found to rise to a Poisson distribution with a mean of 9 if the tennis centre continues to charge its off-peak rate of 6 per court per hour.

What proportion of demand would be turned away if the peak-period price is set at the current off-peak price of 6 per court per hour, and what hourly income would be generated?